Table of Contents
Table of Contents
"Exercises of Basic Analytical Geometry"
INTRODUCTION
CARTESIAN PLANE AND LINE
THE CONICS
"Exercises of Basic Analytical Geometry"
"Exercises of Basic Analytical Geometry"
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
Cartesian plane and translations
line in the Cartesian plane
conics in the Cartesian plane (parabola, circumference, ellipse, hyperbola)
Initial theoretical hints are also presented to make the performance of the exercises understandable
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – CARTESIAN PLANE AND LINE
Cartesian plane
Straight
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercises or 14
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II – THE CONICS
Parable
Circumference
Ellipse
Hyperbole
General considerations on conics
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercises or 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
INTRODUCTION
INTRODUCTION
In this workbook some examples of calculations related to elementary analytic geometry are carried out.
The conceptual evolution of analytic geometry, with respect to normal geometry, is such as to be able to begin a path which, starting from the study of polynomial functions, leads to the graphical resolution of transcendental functions (such as logarithmic, exponential, hyperbolic and trigonometric functions) up to fundamental result of mathematical analysis, i.e. the generalized study of functions of real variable.
In order to understand in more detail what is explained in the resolution of the exercises, the theoretical context of reference is recalled at the beginning of each chapter.
What is exposed in this workbook is generally addressed during the third year of scientific high schools.
I
CARTESIAN PLANE AND LINE
CARTESIAN PLANE AND LINE
Cartesian plane
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Analytic geometry relates the concepts of elementary geometry with the definition of various functions expressible by means of analytic equations.
The functions can be explicit , i.e. take the form y=f(x) or implicit, in the form f(x,y)=0.
The main purpose of analytical geometry is to trace the graph of each type of function to allow a graphical display and to graphically solve the equations, a very powerful mathematical means much more than the simple formal resolution.
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The first approach to analytic geometry assumes Euclidean geometry and the definition of a Euclidean plane for plane geometry and a Euclidean space for solid geometry.
In this manual we will deal only with analytic geometry in the Euclidean plane.
The reference system in this plane is called Cartesian and consists of two oriented straight lines, perpendicular to each other, which are called Cartesian axes .
By convention, the horizontal axis is called the x-axis or x-axis, while the vertical axis is called the y-axis.
Each Cartesian axis is in one-to-one correspondence with the set of real numbers, therefore each point of the Cartesian plane belongs to a set given by the Cartesian product of two sets of real numbers.
Each point can be identified by means of an ordered pair of numbers, the first indicates the abscissa of the point, ie the numerical value resulting from the orthogonal projection on the abscissa axis, the second the ordinate of the point.
All points belonging to the x axis have zero ordinate while all points belonging to the y axis have zero abscissa.
It can be seen that the intersection between the Cartesian axes has Cartesian coordinates given by (0,0): this point is called the origin of the axes .
In analytical geometry, the geometric intersection corresponds to a system of equations, which in turn corresponds to the operation of logical conjunction.
The Cartesian axes divide the plane into four quadrants.
The first quadrant is the one in which both abscissas and ordinates are positive. The other quadrants are numbered following an anti-clockwise count.
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Given a generic point of coordinates (a,b) it is possible to carry out a translation of the Cartesian axes by generating a new reference system.
In this reference system, the new Cartesian coordinates are:
An even function is symmetric about the ordinate axis while an odd function is symmetric about the origin.
From the translation of the axes, it follows that an even function can be symmetrical with respect to one of the straight lines parallel to the ordinate axis and an odd function with respect to one of the points of the Cartesian plane.
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Given two generic points A and B, the distance between them is calculated according to the Pythagorean theorem applied to analytic geometry:
Straight
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A first degree equation identifies a straight line in the Cartesian plane.
The first order implicit function is called the general equation of the straight line and is given by:
If b is different from zero, it is possible to rewrite this equation using the first degree explicit function and, in this case, we speak of a generic equation of a straight line :
In this form, m is called the slope and q ordinate at the origin or intercept .
These denominations derive from geometric considerations: in fact the angular coefficient represents the slope of the line, while the intercept is the value of the ordinate when the line intersects the y axis.
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Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 22.04.2023
ISBN: 978-3-7554-3990-5
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